Once again, forgive me for the necrobumps.
I came across a bunch of "whipped cream" type renders when generating my fractal formulas. One thing that helps is to look at how the polynomial breakdown for complex math plays out. The distributive property groups the polynomials into logical building blocks. The distributive property not only simplifies the strings visually but reduces the cpu complexity, especially if R^2 and I^2 are factored out.
Typically there is a group of terms in the parenthesis followed by a multiplicand of I or R component. Applying the abs() function and or a sign change to these groups individually creates all of the variations of each set. These are interesting fractals in the sense that they can be explored to great depths with minis and minimal distortion. Normally with two manipulatable terms in the real and imaginary components of the formula, this leads to 12 variations for each power.
4th order gets 24 formula instead of 12 due to the symmetry of the fractal resulting in more variations without flips or mirrors. 5th order results in a 4-sided fractal, with additional flips and rotations of existing abs () formula which don't create unique fractals as in 4th. Each fractal has 64 possible outcomes for optionally applying a sign change or abs to each term. Eiminating flips and rotations yeilds 12 possible outcomes, 24 for 4th order. Half of those 64 combinations are asymmetric fractals which exist as chiral pairs. Eliminating mirrors of the asymmetric fractals (example: burning ship with mast above and below the plane) always yields multiples of 3 possible fractal combinations.
When you factor out all of the individual terms in the polynomial, you increase the render time and number of necessary multiplications. You also get the "whipped cream" fractals whenever abs() of sign change is applied individually to one of the terms. I am sure these fractals have depth but the distortion effects are ever changing so one would have to continually adjust the aspect ratio and skew angle throughout the zoom.
The "heart" fractal belongs in the perpendicular family and contains the bottom half of the burning ship flipped about the x axis. Other heart variants contain other fractal's lesser halfs flipped across the axis as well. It is like a deflated Mandelbrot, containing only minis on the needle. KF chose not to add these to his software because they were "uninteresting", a conclusion I made as well.
Good luck with your variations. I believe there are many more interesting "mixed power" polynomial fractals to discover, like the HPDZ Buffalo (first and second order polynomial hybrid). Interestingly the HPDZ buffalo required a non-zero seed to produce minis. Starting with zero looked the same at a glance but you could zoom forever without striking a mini. The HPDZ Buffalo fits the ax^2 + bx^2 + c quadratic mold. There are likely a crapton of interesting polynomial fractals just like this.
Ultrafractal also had a plugin that made some interesting cases. One of them was a hybrid third order formula resembled a 4-bulb Mandelbrot with no elephant valley region, and the Seahorse Valleys had different periods compared to the classic set. Then there is the lamda, a two bulbed Mandelbrot, and others. It had severe limitations however, like complete lack of 64-bit support and render speed went to dinosaur slow once you hit arbitrary.
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September 07, 2017, 04:58:05 PM
